IRWIN. — INVARIANTS OF LINEAR DIFFERENTIAL EXPRESSIONS. 19 



Definition. The weight or total weight of the coefficient 



a pq , p + q = n — k, shall be n — k, and the weight of . pq , 



n — k — (i ' + j) ; the partial weights with respect to x, y of a pq , 



p + q — n — k, shall be p, q respectively, and of { pq , p — i 



and q — j respectively. The weight of a product shall be the sum of 

 the weights of its factors; and a polynomial will be said to be isobaric, 

 totally or partially, if all its terms are of the same weight, total or par- 

 tial. With this definition of weight we have the following proposition : 



Proposition 5. An invariant may or may not be isobaric ; but if 

 not, it is a mere sum of invariants which are isobaric. This statement 

 may be interpreted with respect either to the total or any one of the 

 partial weights. 



We give the proof for the former case. Consider the identity, 

 1(a) = ^I(a). Let the terms of any given weight, w, in 7(a) be 

 represented by G w (a); and let us fix our attention on the correspond- 

 ing terms, G w (a), in 7(a). Suppose we attribute, for our immediate 

 purposes, to ^r the weight zero, to its first derivatives the weight minus 

 one, and so on. Then a comparison of the formulas (17) shows that 

 a pq , p + q = n — k, is of weight n — k in this system of weights, 

 while any of its derivatives is of weight equal to n — k minus the 

 number of differentiations; that is, the weights of the a's and their 

 derivatives are the same as those of the corresponding a's and their 

 derivatives. Thus G w (a) is of weight w, while all the other terms of 

 7(a) are of some other weight; and consequently there can be no can- 

 celling, whole or in part, between those two sets of terms. Therefore, 

 in 7(a) = -yjr^I(a), G w (a) must be equal to the terms of weight w on the 

 right side of the equation : i. e., G w (a) — ^G w {a), as was to be proved. 



This proposition is of service when we are inquiring as to what 

 invariants of a particular degree exist ; in which case we may limit the 

 inquiry to isobaric invariants, since all others can be built up from 

 them by addition. 



§ 5. Particular Invariants. 



A simple set of invariants is furnished by the coefficients of the ad- 

 joint differential expression. That these are invariants follows at once 

 from the proposition: 



Proposition 6. The adjoint of the transformed differential expres- 

 sion, A(77), is yfr times the adjoint of the original expression, L(u). 



